Structure-based determination of equilibrium points of genetic regulatory networks described by differential equation models

A fundamental problem in systems biology consists of determining the equilibrium points of genetic regulatory networks, since the knowledge of these points is often required in order to investigate important properties such as stability. Unfortunately, this problem amounts to computing the solutions of a system of nonlinear equations, and it is well known that this is a difficult problem as no existing method guarantees to find all solutions. This paper addresses this problem for genetic regulatory networks described by differential equation models. By exploiting the structure of these networks, it is shown that one can derive an iterative strategy for progressively singling out the equilibrium points, which does not rely on the solution of any nonconvex optimization problem, and which guarantees to find all equilibrium points. Some numerical examples with small and large sizes (up to 24 state variables) illustrate the benefits of the proposed strategy with respect to existing methods, which often are unable to provide the sought equilibrium points.